Speed Control
3. Speed Control
3.1. Laboratory Objectives
The objective of this laboratory is to develop an understanding of PI control (applied to
speed), how it works, and how it can be tuned to meet required specifications.
In particular you will explore:
Qualitative properties of proportional and integral action.
Set-point weighting.
Design of controllers for specifications on the set-point response.
Integrator windup and windup protection.
Tracking of triangular signals.
Response to load disturbances.
3.2. Preparation And Pre-Requisites
A pre-requisite to this laboratory is to have successfully completed the modelling and
model validation laboratory described in Chapter 2. Before the lab, you should also review
Proportional-Integral (PI) control from your textbook.
From Chapter 2 and for the purpose of speed control, the system can be represented by the
block diagram in Figure 3.1. This block diagram illustrates the parts of the system that are
relevant for speed control.
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Figure 3.1 DCMCT Block Diagram For Speed Control
The process is represented by a block which has motor voltage Vm and torque Td as inputs
and motor speed ωm as the output. The torque is typically a disturbance torque that you
apply manually to the inertial load. The velocity is actually computed in the PIC by filtered
differences of the motor angle θm using the following relationship:
s θm
ωm =
Tf s + 1
where Tf is the filter time constant and s the Laplace operator.
The controller block represents the control algorithm in the computer and the power
amplifier. Vsd is a simulated external disturbance voltage. Make sure that you understand
this fully.
The linear behaviour of the system is described by the transfer functions given in the block
diagram. The major nonlinearities are the saturation of the motor amplifier at 15 V,
Coulomb friction corresponding to 0.5 V, and the quantization of the encoder.
The major unmodeled dynamics is due to the effects of sampling and filtering. The former
can be approximated by a time delay of one sampling period.
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3.3. Introduction: The PI Controller
The PI controller is the most common control algorithm. It is used for a variety of purposes
and it often works very well. For systems with simple dynamics it can give close to optimal
performance and for processes with complicated dynamics, it can often give good
performance provided that specifications are not too demanding. Better performance can,
however, be obtained by using more complicated controllers.
3.3.1. PI Control Law
The linear behaviour of a PI controller can be described by:
t
u( t ) = kp ( b sp r ( t ) − y( t ) ) + ki ⌠
⎮ r ( τ ) − y( τ ) d τ
⌡0
[3.1]
where u(t) is the control signal, r(t) the reference, and y(t) the measured process output. The
reference r(t) is also called the set-point or the command. The linear behaviour of the
controller is governed by three parameters:
kp: proportional gain
ki: integral gain
bsp: set-point weight
Further in this laboratory, we introduce a fourth parameter, aw, which governs the nonlinear
properties of the controller.
Sometimes the filtered measurement yf(t) is also used in the control loop. It is computed
from Tf, the time constant of filter for measured signal. The filter time constant Tf is
typically set to a constant value and it is often combined with the sensor. The filter provides
roll-off at high frequencies. It is important to reduce the effects of sensor noise and it
improves robustness. In this particular case the filtering is incorporated in the calculation of
the velocity from the encoder signal.
3.3.2. The Magic Of Integral Action
A nice property of a controller with integral action is that it always gives the correct steadystate value provided that there is an equilibrium. This can be seen simply by assuming that
there is a steady-state value with constant u(t) = uss, r(t) = rss, and y(t) = yss. Equation [3.1]
can then be written as:
u ss = kp ( b sp rss − y ss ) + ki ( rss − yss ) t
Since the left-hand-side is a constant, the right-hand-side must also be a constant. This
requires that yss = rss. Notice that the only assumption that has been made about the process
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is that there exists an asymptotic steady-state.
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3.4. Nomenclature
The following nomenclature, as described in Table 3.1, is used for the system modelling
and control design.
Symbol
Description
Unit
ωm
Motor speed which can be computed from the motor angle
rad/s
Vm
Voltage from the amplifier which drives the motor
V
Tm
Torque generated by the motor
N.m
Td
Disturbance Torque externally applied to the inertial load
N.m
Vd
Disturbance Voltage corresponding to Td
V
Vsd
Simulated External Disturbance Voltage
V
Im
Motor Armature Current
A
km
Motor Torque Constant
N.m/A
Rm
Motor Armature Resistance
Ω
Jeq
Total Moment Of Inertia Of Motor Rotor And The Load
kg.m2
K
Open-Loop Steady-State Gain
rad/(V.s)
τ
Open-Loop Time Constant
s
s
Laplace Operator
rad/s
h
Sampling Interval
s
t
Continuous Time
s
kp
Proportional Gain
V.s/rad
ki
Integral Gain
V/rad
bsp
Set-Point Weight
aw
Windup Protection Parameter
u
Control Signal
V
r
Reference Signal
rad/s
y
Measured Process Output
rad/s
Tf
Time Constant of filter for measured signal
s
Table 3.1 System Modelling Nomenclature
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3.5. Pre-Laboratory Assignments
3.5.1. PI Controller Design To Given Specifications
In this section you will practice design of a controller for a specified response to reference
signals. One of the key tasks in design is to assess fundamental limitations and assess the
performance that can be achieved. The obtained design will then be verified by an
experimental procedure in Section 3.6.5.
The following problem will be investigated: use the mathematical model of the process to
design a PI controller that gives a step response with:
i) no overshoot.
ii) the following closed-loop undamped natural frequency:
rad ⎤
ω 0 = 16.0 ⎡⎢⎢
⎥
⎣ s ⎥⎦
The relation between motor velocity and motor voltage can be modeled by the following
transfer function:
K
G ω , V( s ) =
[3.2]
τs+ 1
Please answer the following questions.
1. Using the PI control law [3.1] and the process transfer [3.2], determine the closed-loop
block diagram used for speed control. Include the simulated disturbance voltage Vsd.
Solution:
0
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2
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2. Assuming no disturbance, determine the closed-loop transfer function, Gω,r(s), from reference to output, as defined below:
Ω m( s )
G ω , r( s ) =
R( s )
Solution:
0
1
2
3. One possible way to design a controller is to choose controller gains which give a specified characteristic polynomial. One possibility is to choose gains which give the following quadratic characteristic equation:
s2 + 2 ζ ω0 s + ω0
2
[3.3]
Determine the PI controller parameters kp and ki so that the closed-loop system satisfies
the specified characteristic equation [3.3]. That is to say derive kp and ki as functions of
ω0, ζ, K, and τ.
Solution:
0
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4. Choose ζ to get the fastest possible response without overshoot (i.e. critically damped),
as dictated by the design requirements.
Solution:
0
1
2
5. Using kp, ki, and ζ as previously determined and assuming a controller with no set-point
weighting (bsp = 0), express Gω,r(s) under a fully-factored form. Determine the closedloop system time response equation to a unit step input.
Solution:
0
1
2
0
1
2
6. Using the above closed-loop time response equation to a step, the 2% settling time, Ts,
can be expressed as a function of ω0, as shown below:
5.8
Ts =
ω0
Assuming that the closed-loop system meets the design specifications, evaluate its 2%
settling time.
Solution:
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7. Assume that the motor represents the velocity loop in a manufacturing process and that
under normal operating conditions the maximum nominal motor speed ωnom at which the
motor operates is defined as follows:
rad ⎤
ω nom = 150.0 ⎡⎢⎢
⎥⎥
[3.4]
⎣ s ⎦
In order to derive the shortest achievable settling time Ts, calculate the maximum
acceleration, amax, achievable by the motor with the attached inertial load. To do so,
assume that the maximum allowable constant current, Imax, is 0.6 A. Justify this
assumption.
Solution:
0
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8. The motor should switch speed between -ωnom and +ωnom as fast as possible. A simple
estimate of the minimum achievable settling time is the transition time needed to make
that change assuming full acceleration. Calculate the shortest time, Ts_min, to make the
speed transition under the assumption that the motor uses maximum achievable
acceleration.
Solution:
0
1
2
0
1
2
9. Does the designed closed-loop system respect the process physical limitations?
Solution:
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10.Evaluate the PI controller parameters, kp and ki, satisfying the design requirements.
Hint:
Use the nominal values of the process parameters K and τ, as evaluated in Section
2.5.3.6.
Solution:
0
11.In a first design we have used the set-point weight bsp = 0. An alternative design with
faster response will now be investigated. It will be shown that faster response can be
obtained by using a larger value of bsp.
To do this, a PI controller that gives the following transfer function from reference to
output:
ω0
G ω , r( s ) =
[3.6]
s + ω0
will be designed.
Using both proportional and integral gains previously designed, determine and calculate
bsp such that the system closed-loop transfer function is of the form [3.6].
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Solution:
0
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12.Determine the system time response equation to a unit step input. For an asymptotically
stable system, the 2 % settling time, Ts, is defined as the time required for the amplitude
of oscillation to decay permanently to within a 2% margin around the steady-state value.
Using the system closed-loop time response equation to a step and considering that e-4 ≈
0.02, express Ts as a function of ω0.
Solution:
0
1
2
0
1
2
13.Assuming that the closed-loop system meets the design specifications, evaluate its 2 %
settling time. Verify that the controller design with bsp ≠ 0 provides a faster response.
Does it satisfy the system physical limitations?
Solution:
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